We modify the concept of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovász [LLL82] towards a faster reduction algorithm. We organize LLL-reduction in segments of the basis. Our SLLL-bases approximate the successive minima of the lattice in nearly the same way as LLL-bases. For integer lattices of dimension n given by a basis of length 2, SLLL-reduction runs in O(n) bit ope...

The Lovász Local Lemma (LLL) is a powerful tool that can be used to prove that an object having none of a set of bad properties exists, using the probabilistic method. In many applications of the LLL it is also desirable to explicitly construct the combinatorial object. Recently it was shown that this is possible using a randomized algorithm in the full asymmetric LLL setting [R. Moser and G. T...

In a seminal work at EUROCRYPT '96, Coppersmith showed how to nd all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix with ...

Although the LLL algorithm was originally developed for lattice basis reduction, the method can also be used to reduce the condition number of a matrix. In this paper, we propose a pivoted LLL algorithm that further improves the conditioning. Our experimental results demonstrate that this pivoting scheme works well in practice.

The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of “bad” events B in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in B occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms...

Lattices in Computer Science Lecture 2 LLL Algorithm Lecturer: Oded Regev Scribe: Eyal Kaplan In this lecture1 we describe an approximation algorithm to the Shortest Vector Problem (SVP). This algorithm, developed in 1982 by A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovasz, usually called the LLL algorithm, gives a ( 2 √ 3 ) n approximation ratio, where n is the dimension of the lattice. In many...

This paper introduces a number of modifications that allow for significant improvements of parallel LLL reduction. Experiments show that these modifications result in an increase of the speed-up by a factor of more than 1.35 for SVP challenge type lattice bases in comparing the new algorithm with the state-of-the-art parallel LLL algorithm.

The Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid collection ℬ $$ \mathcal{B} “bad” events are mostly independent and have low probability. A seminal algorithm Moser Tardos (J. ACM, 2010, 57, 11) (which we call MT algorithm) gives nearly-automatic randomized algorithms for most constructions based on LLL. However...

In a seminal work at EUROCRYPT ’96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix wit...